Find the area A of the region that is bounded between the curve f(x)=ln(x)+1 and the line g(x)=−4xe+6 over the interval [1,4].

To find the area \( A \) of the region bounded between the curve \( f(x) = \ln(x) + 1 \) and the line \( g(x) = -4x + 6 \) over the interval \([1, 4]\), we follow these steps:

### Step 1: Set up the area formula
The area between two curves \( f(x) \) and \( g(x) \) over the interval \([a, b]\) is given by:
\[
A = \int_a^b \left( f(x) - g(x) \right) dx
\]
Here, \( f(x) = \ln(x) + 1 \) and \( g(x) = -4x + 6 \).

### Step 2: Write the integral
The area \( A \) is:
\[
A = \int_1^4 \left( (\ln(x) + 1) - (-4x + 6) \right) dx
\]

Simplifying the integrand:
\[
(\ln(x) + 1) - (-4x + 6) = \ln(x) + 1 + 

Find the area A of the region that is bounded between the curve f(x)=ln(x)+1 and the line g(x)=-4x+6 over the interval [1,4].

Learn how to calculate the area of the region bounded by the curve f(x)=ln(x)+1 and the line g(x)=-4x+6 over the interval [1,4]. This solution involves setting up a definite integral to find the area between the two curves, a core calculus concept suitable for Grades 11-12 or college-level students.

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